TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Spherical mean-value property for harmonic functions

Statement

If $u \in C^{2} (Ω)$ and $Δ u = 0$ , then $u (x) = M_{u} (x, r)$ whenever $B_{r} (x) ⋐ Ω$ .

Proof

Given: $u$ is classically harmonic and $B_{r} (x) ⋐ Ω$ .

1.1

Radial derivative of a spherical average gives $\frac{d}{d t} M_{u} (x, t) = 0$ for $0 < t \leq r$ [given].

1.2

Hence $M_{u} (x, t)$ is constant, and continuity gives $\lim_{t ↓ 0} M_{u} (x, t) = u (x)$ [given].

2.1

Its value at $r$ is therefore $u (x)$ [step 1.2]. ∎

Depends on

Used by

Ball mean-value property for harmonic functions Corollary

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

PDE source treatment (standard reference, not scraped)